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The symplectic geometry of the three-body problem

Agustin Moreno

TL;DR

The work reframes the spatial CR3BP within modern symplectic and contact geometry, focusing on low-energy near-primary dynamics to connect celestial mechanics with cutting-edge topology. It develops a comprehensive theoretical toolkit—open book decompositions, global hypersurfaces of section, fixed-point theorems for Hamiltonian twist maps, and Floer-theoretic invariants—and then applies these to regularized CR3BP models, including Stark–Zeeman and Levi-Civita regularizations. Key contributions include explicit constructions of adapted open books for the spatial CR3BP, a detailed analysis of return maps in integrable limits like the rotating Kepler problem, and a versatile framework for studying periodic orbits, their bifurcations, and symmetries. The practical impact lies in providing topological and geometric pathways to characterize and design space trajectories using invariant sets, making the CR3BP more amenable to mission-design applications. Overall, the work highlights deep, actionable links between celestial dynamics and modern symplectic topology, illuminating both theory and potential applications in astrodynamics.

Abstract

A book, concerning the classical restricted three body problem, and the approach to this old conundrum coming from the modern methods of symplectic and contact geometry. It is split into Part I (theoretical aspects), and Part II (practical aspects). The main themes are Floer theory, contact topology, symplectic dynamics, and astrodynamics (with a view towards space mission design).

The symplectic geometry of the three-body problem

TL;DR

The work reframes the spatial CR3BP within modern symplectic and contact geometry, focusing on low-energy near-primary dynamics to connect celestial mechanics with cutting-edge topology. It develops a comprehensive theoretical toolkit—open book decompositions, global hypersurfaces of section, fixed-point theorems for Hamiltonian twist maps, and Floer-theoretic invariants—and then applies these to regularized CR3BP models, including Stark–Zeeman and Levi-Civita regularizations. Key contributions include explicit constructions of adapted open books for the spatial CR3BP, a detailed analysis of return maps in integrable limits like the rotating Kepler problem, and a versatile framework for studying periodic orbits, their bifurcations, and symmetries. The practical impact lies in providing topological and geometric pathways to characterize and design space trajectories using invariant sets, making the CR3BP more amenable to mission-design applications. Overall, the work highlights deep, actionable links between celestial dynamics and modern symplectic topology, illuminating both theory and potential applications in astrodynamics.

Abstract

A book, concerning the classical restricted three body problem, and the approach to this old conundrum coming from the modern methods of symplectic and contact geometry. It is split into Part I (theoretical aspects), and Part II (practical aspects). The main themes are Floer theory, contact topology, symplectic dynamics, and astrodynamics (with a view towards space mission design).

Paper Structure

This paper contains 131 sections, 73 theorems, 591 equations, 81 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Organization of the book
  3. Acknowledgments
  4. Theoretical aspects
  5. Basic notions
  6. Symplectic geometry and Hamiltonian dynamics
  7. Symplectic geometry
  8. Hamiltonian dynamics.
  9. Periodic orbits and monodromy.
  10. Symmetries
  11. Monodromy splittings
  12. Compatible almost complex structures
  13. Contact geometry and Reeb dynamics
  14. Contact geometry.
  15. Reeb dynamics
  16. ...and 116 more sections

Key Result

Theorem 1.1.1

If $p\in (M,\omega)$ is an arbitrary point in a symplectic manifold, we can find local charts centered at $p$, so that $(M,\omega)$ is isomorphic to standard phase-space $(\mathbb{R}^{2n},\omega_{std})$ in this local chart.

Figures (81)

  • Figure 1: The fundamental relationship between contact and symplectic geometry is summarized here.
  • Figure 2: Solutions to the Kepler problem are conics.
  • Figure 3: The critical values of $H$.
  • Figure 4: The Hill regions and the Lagrange points for the planar problem.
  • Figure 5: In Moser regularization near the Earth, we add a Legendrian sphere of collisions at the North pole (for fixed energy). The planar problem, which also contains collisions, is an invariant subset.
  • ...and 76 more figures

Theorems & Definitions (152)

  • Example 1.1.1
  • Example 1.1.2
  • Theorem 1.1.1: Darboux's theorem for symplectic manifolds
  • Example 1.1.3
  • Example 1.1.4
  • Remark 1.1.5
  • Theorem 1.1.2: Conservation of energy
  • Definition 1.1.6
  • Theorem 1.1.3
  • Example 1.2.1
  • ...and 142 more